Cognitive load and mixed strategies: On brains and minimax
Sean Du¤yy
J. J. Naddeoz
David Owensx
John Smith{
March 9, 2017
Abstract
It is well-known that laboratory subjects often do not play mixed strategy equilibrium
games according to the equilibrium predictions. However, little is known about the role
of cognition in these strategic settings. We conduct an experiment where subjects play a
repeated hide and seek game against a computer opponent. Subjects play with either fewer
available cognitive resources (under a high cognitive load) or with more available cognitive
resources (under a low cognitive load). Surprisingly, we …nd evidence that subjects under
a high load earn more than subjects under a low load. However, we also …nd that subjects
under a low cognitive load exhibit a greater rate of increase in earnings across rounds,
thus suggesting more learning. Further, while we observe that subjects do not mix in the
predicted proportions and that their actions exhibit serial correlation, we do not …nd strong
evidence these are related to their available cognitive resources. This suggests that the
standard laboratory deviations from equilibrium are not associated with the availability
of cognitive resources. Our results shed light on the extent to which cognitive resources
a¤ect (and do not a¤ect) behavior in games with a mixed strategy equilibrium.
Keywords: bounded rationality, experimental economics, working memory load, cognition,
learning, mixed strategies
JEL: C72, C91
We thank Carlos Alós-Ferrer, Roberto Barbera, Guillaume Fréchette, Johannes Hoelzemann, Hannu Kivimaki, William Neilson, Tibor Neugebauer, Sebastian Olschewski, Rei Sayag, and participants at the Conference
on Economic Design in Istanbul, the 3rd BEEMA conference in Haverford, the 7th Thurgau Experimental Economics Meeting, the SABE/IAREP Conference in Wageningen, and the 2016 London Experimental Workshop.
We thank Lisa Saal for help with the data. John Smith thanks Biblioteca de Catalunya. This research was
supported by Rutgers Research Council Grants #202071, #202167, and #202256.
y
Rutgers University-Camden, Department of Psychology
z
Georgetown University, Department of Economics
x
Haverford College, Department of Economics
{
Corresponding Author; Rutgers University-Camden, Department of Economics, 311 North 5th Street,
Camden, New Jersey, USA 08102; Email: [email protected]; Phone: +1-856-225-6319; Fax: +1-856225-6602
1
1
Introduction
In experimental settings, subjects often do not mix according to the equilibrium predictions.
Existing literature largely …nds that subjects do not mix in the proportions predicted by
equilibrium and that their actions exhibit serial correlation.1 A critique of this literature is
that subjects are often inexperienced in settings where strategic mixing is required. Prompted
by this critique, many studies have examined mixing behavior in settings where decision makers
have ample experience: the …eld.2 Although some deviations are still detected, this literature
mostly …nds that the mixing in …eld settings is closer to the equilibrium predictions than that
found in the laboratory.
In order to better understand the robustness of these deviations from equilibrium predictions, researchers have examined whether experience in mixing in a …eld setting translates
to successfully mixing in a novel experimental setting.3 Our paper is complementary in that
we seek to better understand the observed deviations from the mixed strategy equilibrium
predictions by exploring the role of cognition.
To our knowledge, Geng, Peng, Shachat, and Zhong (2015) is the only other study that
investigates the relationship between cognition and mixing behavior.4 The authors do not …nd
evidence that higher measures of cognitive ability5 are related to behavior consistent with the
equilibrium predictions: proportions of the mixture or serial correlation. Further, Geng et al.
do not …nd a relationship between measures of cognitive ability and earnings in these games.
However, one potential drawback of employing measures of cognitive ability is that these
measures are possibly also correlated with other (observable or unobservable) characteristics
1
See O’Neill (1987), Brown and Rosenthal (1990), Batzilis et al. (2016), Binmore, Swierzbinski, and Proulx
(2001), Geng, Peng, Shachat, and Zhong (2014), Mookherjee and Sopher (1994, 1997), O’Neill (1991), Ochs
(1995), Palacios-Huerta and Volij (2008), Rapoport and Amaldoss (2000, 2004), Rapoport and Boebel (1992),
Rosenthal, Shachat, and Walker (2003), Shachat (2002), Van Essen and Wooders (2015). In fact, Martin et al.
(2014) …nd evidence that chimpanzee subjects are closer to the equilibrium predictions than human subjects.
2
See Azar and Bar-Eli (2011), Bailey and McGarrity (2012), Bar-Eli et al. (2007), Buzzacchi and Pedrini
(2014), Chiappori, Levitt, and Groseclose (2002), Coloma (2007), Emara, Owens, Smith, and Wilmer (2017),
Hsu, Huang, and Tang (2007), Kovash and Levitt (2009), McGarrity and Linnen (2010), Palacios-Huerta
(2003a), Reed, Critch…eld, and Martens (2006), Walker and Wooders (2001).
3
See Palacios-Huerta and Volij (2008), Levitt, List, and Reiley (2010), and Van Essen and Wooders (2015).
4
See Palacios-Huerta et al. (2014) for a study of brain activity during a game with a mixed strategy
equilibrium.
5
Raven’s standard progressive matrices test (Raven and De Lemos, 1990) and a score on a math test.
2
of the subjects, for instance educational opportunities. Thus, these correlations can make the
inferences problematic
We take a complementary approach as we seek to better understand the observed deviations from the equilibrium predictions by manipulating the available cognitive resources of
the subjects. Our study follows other cognitive load experiments that observe behavior or
judgments while the subject has some information committed to memory. This manipulation allows a within-subject design, in the sense that our subjects are placed under di¤erent
cognitive loads, which is not possible with measures of cognitive ability.6
Our subjects are directed to play against computer opponents7 that are programmed to
play one of two exploitative strategies (designed to exploit suboptimal mixing by the subjects)
and one of two naive strategies (designed to allow subjects the possibility of exploiting the
computer). Therefore, subjects will face opponents who are playing only one of a few wellde…ned strategies, and this will facilitate the analysis. Further, using a computer opponent in
an experiment that employs the cognitive load manipulation has the advantage that subjects’
beliefs about the distribution of the cognitive load of the possible opponents and the e¤ects
of the di¤erent cognitive loads on the potential opponents should not a¤ect behavior.
The cognitive load manipulation is designed to diminish the memory capacity of subjects
and to produce a diminished ability to make computations. Since both of these abilities
are important in learning, we aim to determine whether subjects under a high load will
have less success detecting and exploiting naive computer strategies, and have less success
against exploitative computer strategies. In addition, researchers have found that subjects
have di¢ culty detecting and producing random sequences that are required for equilibrium.8
However, the ability to mix in a manner that is consistent with equilibrium would seem to be
6
We note that Carpenter, Graham, and Wolf (2013) …nd that the cognitive load manipulation is more
e¤ective on subjects with a higher measure of cognitive ability. However, Allred, Du¤y, and Smith (2016) do
not …nd such a relationship. Here we …nd some evidence of such a relationship. In particular, we …nd that
higher measures of cognitive ability are associated with better strategic outcomes but that cognitive load can
undo this relationship.
7
Also see Messick (1967), Fox (1972), Coricelli (2005), Levitt, List, and Reiley (2010), Shachat and
Swarthout (2004, 2012), Spiliopoulos (2012, 2013), Samson and Kostyszyn (2015), Shachat, Swarthout, and
Wei (2015), and Bayer and Renou (2016a).
8
For instance, see Wagenaar (1972), Bar-Hillel, and Wagenaar (1991), Rapoport and Budescu (1992),
Budescu and Rapoport (1994), Rabin (2002), and Oskarsson et al. (2009).
3
dependent on the computational ability of the subject. Therefore we seek to determine whether
subjects under a high load will exhibit a mixing proportion further from the equilibrium
prediction and exhibit more serial correlation.
To our surprise, subjects under high cognitive load earned more than subjects under low
load. On the other hand, we …nd that subjects under low load exhibited an increase in
earnings across rounds, whereas we do not …nd such a relationship for subjects under high
load. In addition, we …nd that the response times of subjects under low load are decreasing at
a faster rate than the response times of subjects under high load. We interpret these results as
suggesting that subjects under low cognitive load exhibit a signi…cantly faster rate of learning
than do subjects under high load. Consistent with the previous literature, the behavior in our
experiment exhibits mixture proportions and serial correlation that are inconsistent with the
equilibrium predictions. However, we do not …nd evidence that cognitive load is related to
either the mixing proportions or the observed serial correlation.
The contributions of this paper are as follows. We are the …rst to attempt to better understand mixing behavior by using the cognitive load manipulation. We do not …nd evidence
that the standard results on mixing (suboptimal mixture proportions and serial correlation)
are related to the available cognitive resources of the subject. Our analysis also shows that
subjects with fewer cognitive resources will not necessarily exhibit worse performance, particularly in the early rounds, than will subjects with more cognitive resources. On the other
hand, early round experimentation, which would facilitate learning, can lead to lower payo¤s in these rounds. Our analysis suggests that subjects with more cognitive resources will
exhibit more learning than will subjects with fewer cognitive resources. This is consistent
with the contention that subjects under a low load had the available cognitive resources to
su¢ ciently remember and analyze previous outcomes. Our results shed light on the extent to
which cognitive resources a¤ect (and do not a¤ect) behavior in games with a mixed strategy
equilibrium.
4
2
Related literature
There is a large and growing experimental literature that examines the relationship between
measures of cognitive ability and strategic behavior.9 We take a complementary approach in
that, rather than measure cognitive ability, we manipulate the subjects’ available cognitive
resources.
The cognitive load manipulation is well-studied in nonstrategic settings. Cognitive load
has been found to make subjects more impulsive and less analytical (Hinson, Jameson, and
Whitney, 2003), more risk averse (Whitney, Rinehart, and Hinson, 2008; Benjamin, Brown,
and Shapiro, 2013), more impatient (Benjamin, Brown, and Shapiro, 2013), make more mistakes (Rydval, 2011), exhibit less self control over their actions (Shiv and Fedorikhin, 1999;
Ward and Mann, 2000, Mann and Ward, 2007), fail to process available and relevant information (Gilbert, Pelham, and Krull, 1988; Swann et al., 1990), more susceptible to anchoring
e¤ects (Epley and Gilovich, 2006), perform worse on gambling tasks (Hinson, Jameson, and
Whitney, 2002), perform worse on visual judgment tasks (Morey and Cowan, 2004; Allen,
Baddeley, and Hitch, 2006; Morey and Bieler, 2013; Zokaei, Heider, and Husain, 2014; Allred,
Crawford, Du¤y, and Smith, 2016), a¤ect allocation decisions (Cornelissen, Dewitte, and Warlop, 2011; Schulz et al., 2014),10 a¤ect the evaluations of the fairness of outcomes (van den
Bos et al., 2006), are less dishonest (van’t Veer, Stel, and van Beest, 2014), and are more
in‡uenced by visual salience (Milosavljevic, Navalpakkam, Koch, and Rangel, 2012).11
While many cognitive load studies occur in individual decision settings, only a few involve
strategic settings, and none entail the study of mixing behavior. To our knowledge, studies
of cognitive load in strategic settings only include Milinski and Wedekind (1998), Roch et al.
9
See Al-Ubaydli, Jones, and Weel (2016), Ballinger et al. (2011), Baghestanian and Frey (2016), Bayer
and Renou (2016a,2016b), Benito-Ostolaza, Hernández, and Sanchis-Llopis (2016), Brañas-Garza, Espinosa,
and Rey-Biel (2011), Brañas-Garza, Garcia-Muñoz, and Hernan Gonzalez (2012), Brañas-Garza and Smith
(2016), Burks et al. (2009), Burnham et al. (2009), Carpenter, Graham, and Wolf (2013), Chen, Huang, and
Wang (2013), Corgnet et al. (2016), Coricelli and Nagel (2009), Devetag and Warglien (2003), Fehr and Huck
(2016), Georganas, Healy, and Weber (2015), Gill and Prowse (2016), Grimm and Mengel (2012), Jones (2014),
Jones (2008), Kiss, Rodriguez-Lara, and Rosa-García (2016), Lohse (2016), Palacios-Huerta (2003b), Proto,
Rustichini, and So…anos (2014), Putterman, Tyran, and Kamei (2011), Rydval (2011), Rydval and Ortmann
(2004), and Schnusenberg and Gallo (2011).
10
Although Hauge et al. (2016) does not …nd an e¤ect.
11
Deck and Jahedi (2015) study several e¤ects at a time and …nd that subjects under a cognitive load are
less patient, more risk averse, perform worse on arithmetic tasks, and are more prone to anchoring e¤ects.
5
(2000), Cappelletti, Güth, and Ploner (2011), Carpenter, Graham, and Wolf (2013), Du¤y
and Smith (2014), Buckert, Oechssler, and Schwieren (2014), Allred, Du¤y, and Smith (2016),
and Samson and Kostyszyn (2015).
With the exception of Carpenter et al. (2013) and Samson and Kostyszyn (2015), these
papers describe experiments where the subjects are placed under a cognitive load and play
against a human opponent, who is either under a cognitive load or not. One of the drawbacks
of conducting a cognitive load experiment in a strategic setting with a human opponent is
that the subjects’ beliefs about the distribution of the cognitive load of the opponents and
their beliefs about the e¤ect of the cognitive load on their opponents are not well speci…ed
and are di¢ cult to measure. A design such as ours, which employs a computer opponent, can
address this critique. Further, it allows us to observe the e¤ect of cognitive load on subjects
playing against a small set of distinct varieties of opponent strategies.
3
Experimental design
3.1
Hide and seek game
In our design, subjects play a repeated, deterministic version of the zero-sum, hide and seek
game (Rosenthal, Shachat, and Walker, 2003) against a computer opponent while under an
experimentally manipulated cognitive load. Each subject selected either "Up" or "Down" as
the "Evader" and the computer selected either Up or Down as the "Pursuer." If the computer correctly guesses the subject’s choice then the subject earns 0. The payo¤ to the two
outcomes characterized as successful evasion ({Up,Down} and {Down,Up}) are unequal, with
one yielding a payo¤ of 2 points to the Evader, and the other 1 point. To mitigate concerns
about order or presentation e¤ects, sessions were conducted in which both successful evasion
outcomes yield the higher payo¤. Both of the corresponding payo¤ matrices are presented in
Figure 1. Roughly half of the subjects played the version on the left.12
12
The computer’s actions were presented in red, and the subject’s actions and payo¤s were presented in
blue. A screenshot is available in the Supplemental Online Appendix.
6
Pursuer
Evader
Up
Up
Down
0
1
Pursuer
Evader
Up
Up
Down
0
2
Down
2
0
Down
1
0
Figure 1: Both versions of the hide and seek game, where Evader payo¤s are provided
Each point corresponds to $1:50. To simplify the analysis that follows, we recode the data
from both treatments to correspond to the game on the left, where successfully evading with
Down earns 2. Each subject plays 100 repetitions of the same version of the game. Subjects
receive feedback for that period, including their action, the action of the computer opponent,
and the amount of points earned.
3.2
Memorization task
Before each repetition of the game, a cognitive load was imposed on subjects by directing
them to remember a number. Subjects in the low cognitive load treatment were required to
remember a one-digit number that ranged from 1 to 9. Subjects in the high cognitive load
treatment were required to remember a six-digit number that ranged from 100000 to 999999.
Each number was independently drawn from a uniform distribution on the range provided.
In both treatments, a new number was given for each period. After playing an iteration
of the game and receiving feedback, subjects were asked for the number. Each subject played
50 consecutive repetitions under a high load and 50 consecutive repetitions under a low load.
In this sense, cognitive load was a within-subject manipulation. With probability 0:5 subjects
played …rst under a high load. Subjects were not given feedback about their performance on
the memorization tasks.
3.3
Computer opponent strategies
There are two naive computer strategies. These are strategies that, once detected, can be
exploited in a straightforward manner. One of these naive computer strategies mixes between
Up and Down with equal probability. We refer to this as the Naive 50
7
50 strategy. The
best response to this strategy is to play Down in each period. The other naive computer
strategy mixes with the overall frequency corresponding to the equilibrium prediction, but in
the deterministic pattern of Up-Down-Down-(repeat). We refer to this as the Naive Pattern
strategy. The best response to this strategy is to play Down-Up-Up-(repeat).
There are two exploitative computer strategies. One starts by playing the …rst 5 periods
according to the equilibrium strategy: Up with probability
2
3.
1
3
and Down with probability
Then in the remaining 45 repetitions the computer plays the equilibrium strategy with
probability 0:5 and with probability 0:5 selects the action that would have minimized the
subject’s payo¤s given the proportion of the subject’s previous 4 decisions. We refer to this
as the Exploitative Mix strategy.
The other exploitative strategy also begins playing the equilibrium strategy for the …rst 5
periods, then in the remaining 45 repetitions plays the equilibrium strategy with probability 0:5
and seeks to exploit to the Win-Stay-Lose-Shift tendency13 with probability 0:5. In particular,
if a subject displays behavior consistent with the Win-Stay-Lose-Shift strategy in 2 or 3 of
the previous 3 decisions then the computer selects the action that minimizes the subject’s
payo¤s anticipating the Win-Stay-Lose-Shift strategy. On the other hand, if a subject exhibits
behavior consistent with the Win-Stay-Lose-Shift strategy in 0 or 1 of the previous 3 decisions
then the computer selects the action that minimizes the subject’s payo¤s anticipating the the
Win-Shift-Lose-Stay strategy. We refer to this as the Exploitative WSLS strategy.
Each subject played 50 consecutive rounds against a Naive computer strategy and 50
consecutive rounds against a Exploitative computer strategy. With probability 0:5 subjects
…rst played against a Naive computer opponent.
In order to strike a balance between revealing too little to the subjects and too much
the subjects, we told the subjects the following about the computer strategies. Before the
…rst period, subjects were told, "How does the computer decide what to play?
A number
of possible strategies have been programmed. Some computer strategies can be exploited by
13
See Imhof, Fudenberg, and Nowak (2007), Spiliopoulos (2013), Wang and Xu (2014), and Wang, Xu, and
Zhou (2014).
8
you. Some computer strategies are designed to exploit you. One of these possible strategies
has been selected for the …rst 50 periods." After the …rst 50 periods, subjects were told, "The
computer strategy from the …rst 50 periods is de…nitely not the same as that in the second 50
periods."
3.4
Experimental procedure
At the start of every period, subjects were given 15 seconds to commit a number to memory14
then proceeded to the game. After receiving feedback on the game15 they were asked for the
memorization number. Finally, subjects were given 10 seconds to re‡ect on the number of
periods completed (out of 50) under the computer strategy and cognitive load.16
Each subject earned a $5 show-up fee. Additional payments were designed to decouple the
material incentives from the game in any period with material incentives from the memorization task in that period. Subjects completed 100 repetitions of the game and 100 memorization
tasks. Those who correctly completed all 100 memorization tasks were paid for 30 randomly
selected game outcomes, those who correctly completed 99 were paid for 29, those who correctly completed 98 were paid for 28, and so on, until subjects who correctly completed 70 or
fewer memorization tasks were not paid for any of the game outcomes.
Prior to the incentivized games and memorization tasks, subjects were given an unincentivized test of their understanding of the hide and seek game. Speci…cally, they were asked to
report the number of points that they would earn for all 4 combinations of own actions and
computer actions. They received feedback on their responses. In addition, they were given an
unincentivized opportunity to memorize a six-digit number and an unincentivized opportunity
to memorize a one-digit number. Unlike the incentivized portion of the experiment, subjects
were given feedback about their performance on these memorization tasks.
After completing the incentivized portion of the experiment, subjects reported their gender,
whether they were an economics major, whether they have taken a game theory course, an
14
Subjects could click to proceed to the next stage but after 15 seconds would proceed automatically.
Subjects were given 20 seconds to re‡ect on the game feedback. They could click to proceed to the next
stage but after 20 seconds would proceed automatically.
16
Subjects could click to proceed to the next period but would automatically do so after 10 seconds.
15
9
optional estimate of their grade point average17 (GPA), and a rating of the di¢ culty in recalling
the large and the small memorization numbers. These di¢ culty ratings were elicited on a
scale of 1 ("Very Di¢ cult") to 7 ("Not Very Di¢ cult"). After these questions were completed,
subjects learned their earnings. Subsequently the experimenter took an image of the right
hand of the subjects with a digital scanner18 then they were paid in cash.
A total of 130 subjects participated in the experiment. The experiment was programmed
and conducted with the software z-Tree (Fischbacher, 2007).19 Of the 130 subjects, 78 were
students at Rutgers University-Camden and 52 were students at Haverford College.20 There
were 13 sessions conducted at Camden and 3 at Haverford. None of the 16 sessions lasted
longer than 60 minutes and the average subject earned $33.
3.5
Discussion of the experimental design
We employ only 4 strategies for the computer opponent. As a result, we can compare strategic behavior against a few well-de…ned opponent strategies. In the case of human opponents,
the number of these strategies would either not be known or they would not be well-de…ned.
Further, we allude to the naive and exploitative strategies in the description to the subjects
because these communicate the range of possible computer strategies. Moreover, this description increases the plausibility of the claim that the computer strategy would not be identical
in both blocks of 50.
In the exploitative computer strategies, the computer executes exploitation with probability 0:5, rather than with certainty. A deterministic strategy could be detected by some
subjects, rendering this strategy exploitable.
Given the exploitative strategies, the reader might ask why the subject should mix. The
answer, applicable to all possibly exploitative strategies of the opponent, is that mixing according to minimax is the only way to avoid the risk of being exploited. Further, any concerns
about incentives to mix against our exploitative strategies also apply to settings with human
17
Grade point average ranges from 0:0 to 4:0, and is increasing in performance.
We employed a Canon CanoScan 4507B002 LiDE110 Color Image Scanner. We report the analysis of this
data in Du¤y et al. (2017).
19
The z-Tree code is available from the corresponding author upon request.
20
The Haverford subjects were recruited via ORSEE (Greiner, 2015).
18
10
opponents.
Recall that subjects memorize a di¤erent number every period, keeping cognitive load
relatively constant across periods. On the other hand, memorizing only a single number
across several periods could produce a non-constant load, as the number could be rehearsed
across periods. It is not known whether this taxing of cognitive resources would increase or
decrease across periods, however it would likely not be constant.
Computer strategies, both naive and exploitative, are only vaguely described to subjects
as "...can be exploited by you..." and "...designed to exploit you..." We acknowledge that the
meaning of these sentences may be ambiguous to some subjects. However, it is important
to the design of the experiment that subjects attempt to discern a strategy on the part of
the computer, and direct cognitive resources towards this goal. Our language clari…es that
a motivation can be discerned. Language that more precisely speci…ed the four strategies
has two potential drawbacks. First, anything speci…c about the strategies themselves would
render all four of them too easy to exploit. Second, wording that pins down the computer’s
game-theoretic motivation21 could exacerbate di¤erences between subjects who are familiar
with Game Theory and Economics, and those who are not. Our description was selected
because it clari…es that subjects should try to learn something about the computer opponent
without being speci…c about its precise nature.
The goals of our incentive scheme are as follows: strongly incentive the memorization task,
keep incentives for memorization in each period independent from incentives for the game
decision in that particular period, and maintain identical incentives for high and low load
memorization periods. Our solution to this was to not provide feedback on the memorization
task and to pay a number of randomly selected game outcomes that is decreasing in the
number of incorrect memorization tasks. Only 1 subject out of 130 failed to correctly perform
at least 70 memorization tasks, suggesting that the incentive scheme was properly calibrated.
In addition, as feedback was not given on the memorization task, it is not clear whether
subjects realized that they were near or below 70 correct. Finally, while incorrectly answering
21
For example, the instructions could have explained "Some computer opponents are programmed to detect
and best-respond to your strategy."
11
a memorization task decreases incentives, this a¤ects high and low load trials equally and we
are primarily interested in the di¤erence between these treatments.
4
Results
4.1
Summary statistics
We begin with the summary statistics of the variables of interest. Correct is a dummy variable
indicating that the memorization task was correctly completed, Down is a dummy indicating
that the Down action was selected, and Earnings is the amount earned in a particular game
outcome: 0, 1, or 2 points. Female, Economics, and Game Theory are dummies indicating
gender, that the subject was an economics major and that the subject reported having taken
a game theory course. GPA refers to the subject’s self-reported grade point average. Table 1
lists the means of these variables, while Table 2 reports the Spearman correlation coe¢ cients.
Table 1: Summary statistics
Mean
Correct
0:929
Down
0:555
Earnings
0:733
Female
0:531
Economics
0:169
Game Theory
0:177
GPA (optional)
3:365
Correct, Down, and Earnings have 13,000 observations. Female, Economics,
and Game Theory have 130 observations. GPA has 103 observations.
Table 2: Spearman non-parametric correlation coe¢ cients
1
2
3
4
1 Correct
1:00
2 Down
0:0110
1:00
3 Earnings
0:0075
0:0343
1:00
4 Female
0:0142
0:0044
0:0228
1:00
5 Economics
0:0083
0:0050
0:0049
0:316
6 Game Theory
0:0073
0:0180
0:0122
0:210
7 GPA (optional) 0:0767
0:0066
0:0254
0:0767
12
5
6
1:00
0:436
0:234
1:00
0:0649
Each correlation between variables 1, 2, or 3, and variables 4, 5, or 6 has
13,000 observations. Each correlation between variables 1, 2, or 3, and variable
7 has 10,3000 observations. Each correlation between variables 4, 5, or 6, and
variable 7 has 103 observations. Each correlation among variables 1, 2, and 3
has 13; 000 observations. Each correlation among variables 4, 5, and 6 has 130
observations.
denotes p < 0:001,
denotes p < 0:01, denotes p < 0:05, and
y denotes p < 0:1.
We observe that higher GPA subjects tend to earn more and are more likely to correctly
perform the memorization task. While we do not observe a relationship between Earnings and
either Economics or Game Theory, we observe a negative relationship between Earnings and
Female. Finally, we do not observe a correlation between earnings and either Economics or
Game Theory. This suggests that our description of the computer strategies did not advantage
those with exposure to the study of games.22
We note that the high load memorization tasks were correct (87:97%, 5718 of 6500) with a
signi…cantly lower frequency than the low load memorization tasks (97:88%, 6362 of 6500), according to a Mann-Whitney test Z =
22:03, p < 0:001. This suggests that our manipulation
was successful23 although it should be noted that the success rate was relatively high in the
high load treatment. As each of the 130 subjects attempted 50 high load memorization tasks
and 50 low load memorization tasks, Table 3 presents a characterization of the subject-level
distribution of number of correct memorization tasks by cognitive load treatment and the
number pooled across treatments.
Table 3: Distribution of subjects by number of correct memorization tasks
Within blocks of 50
46 50
41 45 36 40 31 35 26 30 23 25
High load
72
30
13
10
2
3
Low load
125
5
0
0
0
0
96
Pooled
100
64
91
95
31
Across both blocks of 50
86 90 81 85 76 80
16
11
5
71
75
2
< 23
0
0
Total
130
130
< 71
1
Total
130
22
Apparently being an economics major is not good for one’s GPA: we note a negative relationship between
GPA and Economics.
23
As a robustness check, we run a repeated measures logistic regression with Correct as the dependent
variable and High load as the independent variable. The High load estimate is negative and signi…cant, t =
13:56, p < 0:001.
13
The upper panel characterizes the subject-level distribution of number of correct memorization tasks by cognitive load treatment. The lower panel characterizes
the subject-level distribution of the correct memorization tasks across both cognitive load treatments.
Table 3 shows that 111 of the 130 subjects successfully completed more than 85% of
their memorization tasks correctly. This suggests that the incentives were su¢ cient to elicit
cognitive e¤ort on these tasks.
4.2
Earnings di¤erences
Now we examine di¤erence in earnings by cognitive load treatment. In each of the regressions below we include dummy variables identifying the computer opponent treatment and
the interaction of these dummies with the high load dummy. In addition, we consider speci…cations that account for the repeated nature of the observations. In these repeated measures
regressions, we estimate an exchangeable covariance matrix, clustered by subject. In other
words, we assume a unique correlation between any two observations involving a particular
subject. However, we assume that observations involving two di¤erent subjects are statistically independent. We also consider speci…cations that control for Female, Economics, and
Game Theory. We refer to this collection of variables as Demographics. We also account for
self-reported GPA. Recall that a response to GPA was optional and only 103 of 130 subjects
provided a response. This analysis is summarized in Table 4.
Table 4: Regressions of earnings
(1)
(2)
High load
0:0626
0:0692
(0:0284) (0:0327)
GPA
(3)
0:0791
(0:0333)
(4)
0:0968
(0:0369)
0:0457
(0:0229)
Y es
Y es
Y es
31212:3
13,000
Y es
Y es
Y es
24767:3
10,300
GPA*High load
Strategy dummies
Repeated measures
Demographics
AIC
Observations
Y es
No
No
31221:7
13,000
Y es
Y es
No
31199:0
13,000
14
(5)
0:4220
(0:1240)
0:0926
(0:0285)
0:0971
(0:0354)
Y es
Y es
Y es
24764:7
10,300
The repeated measures regressions estimate an exchangeable covariance matrix, clustered by subject. We do not provide the estimates of the intercepts, the
individual demographics variables, the covariance estimates, or the strategy dummies. AIC refers to the Akaike information criterion (Akaike, 1974).
denotes
p < 0:001, denotes p < 0:01, denotes p < 0:05, and y denotes p < 0:1.
In each speci…cation, we …nd that subjects under a high cognitive load earn a signi…cantly
larger amount than subjects under a low load. We also …nd a positive relationship between
GPA and earnings. However, the estimate of the GPA-High load interaction suggests the
positive relationship between GPA and earnings is driven by subjects under a low load.24;25
To better understand the nature of the di¤erences in earnings summarized in Table 4, we
provide the estimates of the Least Square Means involving the strategy dummy variables from
each of the 5 regressions. In other words, we list the predicted population averages within
each strategy and cognitive load treatment. In addition, we test for the di¤erences in these
estimates by cognitive load treatment. We summarize this analysis in Table 5.
24
The reader might worry about whether high cognitive load in the …rst block of 50 a¤ects behavior in the
second block. We therefore supplemented the regressions in Table 4 with a First block dummy variable and
its interaction with High load. We …nd that First block dummy is positive and signi…cant at 0.05, and we …nd
that the interaction is not signi…cant. This perhaps indicates that subjects experienced fatigue in the second
block, regardless of the load in the …rst block.
25
In regressions (3) (5) we note that neither the Economics nor the Game Theory variables are signi…cant.
This suggests that our description of the computer strategies did not advantage those with an exposure to the
study of games.
15
Table 5: Estimated Least Square Means of
(1)
Naive 50-50 High load
0:779
Naive 50-50 Low load
0:794
Naive Pattern High load
0:855
Naive Pattern Low load
0:753
Exploitative WSLS High load
0:707
Exploitative WSLS Low load
0:735
Exploitative Mix High load
0:664
Exploitative Mix Low load
0:602
Naive 50-50 Di¤erence
0:0153
(0:0274)
Naive Pattern Di¤erence
0:1012
(0:0303)
Exploitative WSLS Di¤erence
0:0278
(0:0287)
Exploitative Mix Di¤erence
0:0626
(0:0284)
the strategy dummies from Table 4
(2)
(3)
(4)
(5)
0:783
0:782
0:801
0:794
0:792
0:795
0:796
0:799
0:848
0:840
0:865
0:877
0:756
0:758
0:748
0:745
0:701
0:702
0:691
0:697
0:736
0:733
0:722
0:727
0:670
0:674
0:681
0:675
0:601
0:595
0:584
0:579
0:0086
0:0137 0:00516
0:0052
(0:0316) (0:0318) (0:0352) (0:0353)
0:0926
0:0822
0:1177
0:1327
(0:0348) (0:0357) (0:0398) (0:0401)
0:0346
0:0304
0:0311
0:0300
(0:0331) (0:0333) (0:0376) (0:0375)
0:0692
0:0791
0:0968
0:0954
(0:0327) (0:0333) (0:0369) (0:0368)
We provide the Least Square Means of the strategy dummy variables for each of
the 5 regressions summarized in Table 4. We also provide the results of a Wald test
of the di¤erences of these Least Square Means between cognitive load treatments.
denotes p < 0:001,
denotes p < 0:01, denotes p < 0:05, and y denotes
p < 0:1.
We note that earnings can vary among strategy and cognitive load treatments. However,
the only signi…cant di¤erences between the cognitive load treatments occur in the Naive Pattern and the Exploitative Mix treatments. In both of these strategy treatments, we …nd that
the subjects under a high cognitive load have higher earnings than subjects under a low load.
4.3
Earnings di¤erences over time
To better understand subjects’ higher earnings under high cognitive load, we now consider
the trajectory of earnings. We de…ne Round to be the number of periods under a particular
computer opponent treatment and cognitive load treatment. Therefore, Round ranges from
1 to 50. Other than the inclusion of the Round variable and the interaction of Round and
cognitive load treatment, the analysis is identical to that summarized in Table 4. This analysis
is summarized in Table 6.
16
Table 6: Regressions of earnings
(1)
High load
0:114
(0:038)
Round
0:0019
(0:0007)
Round*High load
0:0020
(0:00098)
GPA
across rounds
(2)
(3)
0:120
0:130
(0:041)
(0:042)
0:0019
0:0019
(0:0007)
(0:0007)
0:0020
0:0020
(0:00097) (0:00097)
(4)
0:146
(0:046)
0:0020
(0:0008)
0:0019y
(0:0011)
0:046
(0:023)
GPA*High load
Strategy dummies
Repeated measures
Demographics
AIC
Observations
Y es
No
No
31239:6
13,000
Y es
Y es
No
31216:8
13,000
Y es
Y es
Y es
31230:1
13,000
Y es
Y es
Y es
24785:8
10,300
(5)
0:471
(0:127)
0:0020
(0:0008)
0:0019y
(0:0011)
0:093
(0:029)
0:097
(0:035)
Y es
Y es
Y es
24783:1
10,300
The repeated measures regressions estimate an exchangeable covariance matrix, clustered by subject. We do not provide the estimates of the intercepts, the
individual demographics variables, the covariance estimates, or the strategy dummies. AIC refers to the Akaike information criterion (Akaike, 1974).
denotes
p < 0:001, denotes p < 0:01, denotes p < 0:05, and y denotes p < 0:1.
The positive and signi…cant high load coe¢ cient suggests that subjects under a high load
earn more in the early rounds. However, we observe a positive and signi…cant Round coef…cient in addition to a negative and signi…cant Round-High load interaction. This indicates
that the subjects under a low load exhibit an improved earnings across rounds, however, the
subjects under a high load do not exhibit such an improvement.26 This …nding is robust to
the speci…cation of the analysis. Our analysis suggests that while subjects under a low load
earn less than subjects under a high load in the early rounds, the subjects under a low load
appear to be catching up. This result is consistent with the interpretation that subjects under
a low load exhibit a greater learning across rounds. We also note that our results involving
GPA are similar to that found in Table 4. We …nd that there is a relationship between higher
GPA and earnings, but only for subjects under a low load.
26
In an analysis similar to that in Table 6, we note that the GPA-Round interaction, the GPA-Round-High
load interaction, and the Strategy dummies-High load interactions do not yield helpful insights. Further, in
the Supplemental online appendix we present the results of regression (2) in Table 6, restricted to computer
opponent treatments.
17
4.4
Di¤erences in response time
Research …nds a positive relationship between the time spent deciding on a choice and the
di¢ culty of the choice.27 In other words, decisions where one option is clearly better than
the others tends to take less time than decisions where this is not the case. In order to
better understand the analysis of earnings across rounds, here we study the response times
across rounds. We run the analogous regressions as summarized in Table 6 but we employ
the response time of the game decision as the dependent variable. Therefore, a higher value
indicates more time spent on the decision. As response times are constrained to be nonnegative, we perform the analysis by taking the natural log of the response times. We note that
the response times ranged from 0 to 29. In order to avoid taking the log of 0, we add 1 to all
values.28 This analysis is summarized in Table 7.
Table 7: Regressions of the natural
(1)
High load
0:2135
(0:0223)
Round
0:0084
(0:0004)
Round*High load
0:0017
(0:0006)
GPA
log of response times across rounds
(2)
(3)
(4)
0:2581
0:2345
0:2232
(0:0484)
(0:0495)
(0:0570)
0:0084
0:0084
0:0075
(0:0004)
(0:0004)
(0:0004)
0:0017
0:0017
0:0007
(0:0005)
(0:0005)
(0:0006)
0:0200
(0:0549)
GPA*High load
Strategy dummies
Repeated measures
Demographics
AIC
Observations
Y es
No
No
17530:9
13,000
Y es
Y es
No
14154:4
13,000
Y es
Y es
Y es
14160:8
13,000
Y es
Y es
Y es
11212:3
10,300
(5)
0:3420
(0:0838)
0:0075
(0:0004)
0:0007
(0:0006)
0:0032
(0:0557)
0:0355y
(0:0183)
Y es
Y es
Y es
11214:7
10,300
The repeated measures regressions estimate an exchangeable covariance matrix, clustered by subject. We do not provide the estimates of the intercepts, the
individual demographics variables, the covariance estimates, or the strategy dummies. AIC refers to the Akaike information criterion (Akaike, 1974).
denotes
p < 0:001, denotes p < 0:01, denotes p < 0:05, and y denotes p < 0:1.
27
See Wilcox (1993), Mo¤att (2005), Rubinstein (2007), Alós-Ferrer, Granić, Shi, and Wagner (2012), Chen
and Fischbacher (2015), and Alós-Ferrer, Granić, Kern, and Wagner (2016).
28
There were 3452 observations with a response time of 0.
18
Table 7 provides evidence that subjects under a high load take less time to reach their
decisions. This is possibly done in an e¤ort to quickly proceed to the memorization task. We
also observe that the response time decreases across rounds for subjects under a low load. In
addition, the Round-High load interaction suggests that subjects under a low load exhibit a
greater increase in the decision speed across rounds than subjects under a high load. This is
consistent with the interpretation of the analysis summarized in Table 6 that subjects under
a low load exhibit a greater amount of learning than subjects under a high load. However,
we note that this result is not robust to speci…cations that include GPA. Interestingly, we
do not …nd a relationship between response time and GPA, analogous to that found in Table
6. On the other hand, we …nd some evidence that higher GPA subjects under a high load
exhibit less rapid decisions.29 Overall the analysis summarized in Table 7 is consistent with
the contention that subjects under a low load are learning the strategy of the opponent better
than subjects under a high load.
4.5
Mixture proportions
We now test whether the subjects mixed in the proportions as predicted by equilibrium: Up
with probability 0:6667 and Down with probability 0:3333. Here we restrict attention to
observations against Exploitative computer strategies. This is because there are di¢ culties
interpreting the mixing proportions against the Naive computer strategies.
We conduct a binomial
2
test on each subject.30;31 Performing a joint test on the 76
subjects under a high load by summing their test statistics, we reject the hypothesis that, on
aggregate, they mix with these proportions,
a joint binomial
2
2 (76; 1)
= 1026:22, p < 0:001. We also conduct
test on 53 the subjects under a low load by summing their test statistics,
and again we reject the hypothesis that they mix with the Nash equilibrium proportions,
2 (53; 1)
= 774:08, p < 0:001.
29
We note that the linear regressions for response times are qualitatively similar to those summarized in
Table 7. This analysis is available from the corresponding author upon request.
30
See the Supplemental Online Appendix for the subject-level data. Note that one subject selected Down
in every period and therefore we cannot perform a binomial 2 test on this subject.
31
We note that there does not exist a signi…cant Spearman correlation between the 2 statistic and the
Female, Game Theory, Economics, and GPA variables.
19
Next we test the hypothesis that the subjects under high and low load are signi…cantly
di¤erent in this regard. We conduct a two-sample Kolmogorov-Smirnov test32 on the distribution of the individual
2
test statistics and …nd that they are not signi…cantly di¤erent,
K = 0:164, p = 0:37. This qualitative result is not changed when we restrict attention to
the last 25 periods of each 50 period block, K = 0:129, p = 0:70. We also test for di¤erences
between the treatments using a Mann-Whitney test on the percentage of Down actions against
an exploitative computer opponent. We …nd that the subjects under a high load (54:05%)
had a signi…cantly di¤erent mixture than subjects under a low load (56:44%), Z = 1:910,
p = 0:056. However, the di¤erence between subjects under high load (53:42%) and low load
(56:30%) is not signi…cant when we restrict attention to the …nal 25 periods of the 50 period
block, Z = 1:622, p = 0:105.
Therefore, consistent with the previous literature, we …nd that the subjects do not mix
in the equilibrium proportions. However, we do not …nd strong evidence of a signi…cant
di¤erence between the mixture proportions of subjects under a high load and subjects under
a low load. This suggests that the availability of cognitive resources is not related to the
observed deviations from the equilibrium mixture proportions in this game.
4.6
Serial correlation
Next we investigate whether the actions in our data exhibit serial correlation. As in the
previous subsection, we restrict attention to observations against an Exploitative opponent.
This is because exhibiting a mixture with serial correlation against a Naive opponent is a best
response.
In order to detect serial correlation, we perform tests of runs, as described in Gibbons and
Chakraborti (1992). A run (r) is de…ned to be a sequence of one or more identical actions
followed by a di¤erent action or no action at all. Given the number of Up actions (nU ) and the
number of Down actions (nD ) selected by a subject, we are able to calculate the probability
of observing any feasible number of runs.33 For every subject, given nD and nU we calculate
32
33
See Gibbons and Chakraborti (1992).
Given nU > 0 and nD > 0, there must be at least 2 runs and the maximum possible number of runs is
20
the probability density function of the number of runs:
f (rjnU ; nD ) =
8
>
>
>
>
>
>
<
n
nD 1
1
1 (r
2) 1
nD +nU
nU
2(( rU)
2
)(
(
)
)
if r is even
>
>
n
1 n
1
n
1 n
1
>
>
( rU 1 )( rD 3 )+( rU 3 )( rD 1 )
>
>
2
2
2
2
:
if r is odd.
U)
(nDn+n
U
From the density function, we can calculate the cumulative distribution function:
F (rjnU ; nD ) =
r
X
f (kjnU ; nD );
k=1
which is the probability of observing r or fewer runs. Similar to Walker and Wooders (2001),
Palacios-Huerta and Volij (2008), and Levitt, List, and Reiley (2010), we calculate two statistics, F (r
1jnU ; nD ) and F (rjnU ; nD ), for each subject.34 At a 5% level of signi…cance,
we would reject the null hypothesis of independence, if either F (rjnU ; nD ) < 0:025 or if
1
F (r
1jnU ; nD ) < 0:025. Because we plan to run one-sample Kolmogorov-Smirnov tests
on these probabilities, as Walker and Wooders (2001), for each subject we take a draw from
the uniform distribution with F (rjnU ; nD ) as the upper bound and F (r
1jnU ; nD ) as the
lower bound. This leaves us with a single probability estimate for each subject.35 If the actions
are selected independently then these probabilities would be distributed as a uniform between
0 and 1.
We perform a one-sample Kolmogorov-Smirnov test that the 53 probabilities associated
with subjects under a low load are uniformly distributed between 0 and 1. We reject the
hypothesis that the probabilities are distributed as a uniform, K = 0:174, p = 0:071. Figure 2
illustrates the test on subjects under a low load. We also perform a one-sample KolmogorovSmirnov test that the 76 probabilities associated with subjects under a high load are uniformly
distributed between 0 and 1. Again, we reject the hypothesis, K = 0:246, p < 0:001. Figure
3 illustrates the test on the subjects under a high load.
equal to 2 min(nU ; nD ) + 1.
34
See the Supplemental Online Appendix for the subject-level data.
35
We note that there does not exist a signi…cant Spearman correlation between these probability estimates,
and the Female, Game Theory, Economics, and GPA variables.
21
<<Figures 2 and 3 about here>>
While subjects in neither cognitive load treatments appear to be mixing in an independent
fashion, it remains to be seen whether the distribution associated with subjects under a high
load is di¤erent from the distribution associated with subjects under a low load. We perform a
two-sample Kolmogorov-Smirnov test and we fail to reject the hypothesis that the distributions
are di¤erent, K = 0:158, p = 0:42. This qualitative result remains unchanged when we restrict
the analysis to the …nal 25 periods in the 50 period block, K = 0:091, p = 0:97.
5
Discussion
The experimental literature largely …nds that subjects do not mix in the proportions predicted
by equilibrium and that actions exhibit serial correlation. We …nd these features in our data,
however we do not …nd evidence that they are related to cognitive load. Therefore, we do
not …nd evidence that these standard experimental results on mixing are associated with the
available cognitive resources of the subject.
These results are reminiscent of the …ndings reported in Geng et al. (2015). These authors
do not …nd a relationship between measures of cognitive ability and either the mixture proportions or the serial correlation of actions. Although the design of Geng et al. (2015) exhibits
(adolescent subjects, human opponents, measures of cognitive ability) notable di¤erences from
our design (college student subjects, computer opponents, cognitive load manipulation), neither study …nds a relationship between cognition and either mixing proportions or serially
correlated actions.
We also …nd surprising evidence that subjects under a high cognitive load earn more than
subjects under a low cognitive load, particularly in the early rounds. This is consistent with
the explanation that subjects under a high load are employing a simple, stable strategy and
subjects under a low load are engaging in experimentation during those early rounds.
In addition, we …nd that subjects under a low cognitive load exhibit increases in earnings
across rounds, while those under a high cognitive load do not.36 An interpretation of this
36
Geng et al. (2015) did not study the trajectory of earnings across rounds.
22
result is that the subjects under a low load exhibit more learning than subjects under high
load. Our analysis of the response times is also consistent with this interpretation. This result
has an intuitive appeal because remembering and analyzing previous outcomes would seem to
require available cognitive resources.
Gill and Prowse (2016) …nd a similar result, albeit in a di¤erent setting. These authors
observe that subjects with higher measured cognitive ability exhibit a faster convergence to
the equilibrium prediction in a repeated beauty contest game.
Whereas our results shed light on the role of cognition a¤ecting (and not a¤ecting) behavior
in games with a mixed strategy equilibrium, we acknowledge that there is much work to be
done on the topic. We leave it to future research to determine whether there is a su¢ cient
number of rounds where subjects under a low load would earn more than subjects under a high
load. Also, it is possible that subjects play a computer opponent di¤erently than a human
opponent because the computer might be expected to employ a more stable strategy. Further,
our design allows the subjects time between feedback and the choice in the game that is not
loaded. We are interested to know the e¤ects of imposing a cognitive load during this time.
These and other interesting questions are a matter for future research.
23
References
Akaike, Hirotugu (1974): "A new look at the statistical model identi…cation," IEEE Transactions on Automatic Control, 19(6), 716–723.
Allen, Richard J., Baddeley, Alan D., and Hitch, Graham J. (2006): "Is the binding of
visual features in working memory resource-demanding?" Journal of Experimental Psychology:
General, 135(2), 298–313.
Allred, Sarah, Crawford, L. Elizabeth, Du¤y, Sean, and Smith, John (2016): "Working
memory and spatial judgments: Cognitive load increases the central tendency bias," Psychonomic Bulletin and Review, 23(6), 1825–1831.
Allred, Sarah, Du¤y, Sean, and Smith, John (2016): "Cognitive load and strategic sophistication," Journal of Economic Behavior and Organization, 125, 162–178.
Alós-Ferrer, Carlos, Granić, Ðura-Georg, Kern, Johannes, and Wagner, Alexander K.
(2016): "Preference Reversals: Time and Again," Journal of Risk and Uncertainty, 52(1),
65–97.
Alós-Ferrer, Carlos, Granić, Ðura-Georg, Shi, Fei, and Wagner, Alexander K. (2012):
"Choices and preferences: Evidence from implicit choices and response times," Journal of
Experimental Social Psychology, 48(6), 1336–1342.
Al-Ubaydli, Omar, Jones, Garett, and Weel, Jaap (2016): "Average player traits as predictors of cooperation in a repeated prisoner’s dilemma," Journal of Behavioral and Experimental
Economics, 64, 50–60.
Azar, Ofer H. and Bar-Eli, Michael (2011): "Do soccer players play the mixed-strategy
Nash equilibrium?" Applied Economics, 43(25), 3591–3601.
Baghestanian, Sascha and Frey, Seth (2016): "GO Figure: Analytic and Strategic Skills
are Separable," Journal of Behavioral and Experimental Economics, 64, 71–80.
Bailey, Brett James and McGarrity, Joseph P. (2012): "The E¤ect of Pressure on MixedStrategy Play in Tennis: The E¤ect of Court Surface on Service Decisions," International
Journal of Business and Social Science, 3(20), 11–18.
Ballinger, T. Parker, Hudson, Eric, Karkoviata, Leonie, and Wilcox, Nathaniel T. (2011):
"Saving behavior and cognitive abilities," Experimental Economics, 14, 349–374.
Bar-Eli, Michael, Azar, Ofer H., Ritov, Ilana, Keidar-Levin, Yael, and Schein, Galit (2007):
"Action bias among elite soccer goalkeepers: The case of penalty kicks," Journal of Economic
Psychology, 28(5), 606–621.
Bar-Hillel, Maya, and Wagenaar, Willem A. (1991): "The perception of randomness,"
Advances in Applied Mathematics, 12(4), 428–454.
24
Batzilis, Dimitris, Ja¤e, Sonia, Levitt, Steven, List, John A., and Picel, Je¤rey (2016):
"How Facebook Can Deepen our Understanding of Behavior in Strategic Settings: Evidence
from a Million Rock-Paper-Scissors Games," working paper, Harvard University.
Bayer, Ralph and Renou, Ludovic (2016a): "Logical omniscience at the laboratory," Journal of Behavioral and Experimental Economics, 64, 41–49.
Bayer, Ralph and Renou, Ludovic (2016b): "Logical abilities and behavior in strategic-form
games," Journal of Economic Psychology, 56, 39–59.
Benito-Ostolaza, Juan M., Hernández, Penélope, and Sanchis-Llopis, Juan A. (2016): "Are
individuals with higher cognitive ability expected to play more strategically?" Journal of
Behavioral and Experimental Economics, 64, 5–11.
Binmore, Ken, Swierzbinski, Joe, and Proulx, Chris (2001): "Does minimax work? An
experimental study," Economic Journal, 111(473), 445–464.
Brañas-Garza, Pablo, Espinosa, Maria Paz, and Rey-Biel, Pedro (2011): "Travelers’types,"
Journal of Economic Behavior and Organization, 78, 25–36.
Brañas-Garza, Pablo, Garcia-Muñoz, Teresa and Hernan Gonzalez, Roberto (2012): "Cognitive e¤ort in the Beauty Contest Game," Journal of Economic Behavior and Organization,
83(2), 254–260.
Brañas-Garza, Pablo and Smith, John (2016): "Cognitive abilities and economic behavior,"
Journal of Behavioral and Experimental Economics, 64, 1–4.
Brown, James N. and Rosenthal, Robert W. (1990): "Testing the minimax hypothesis: a
re-examination of O’Neill’s game experiment," Econometrica, 58(5), 1065–1081.
Buckert, Magdalena, Oechssler, Jörg, and Schwieren, Christiane (2014): "Imitation under
stress," working paper, University of Heidelberg.
Budescu, David V. and Rapoport, Amnon (1994): "Subjective randomization in one-and
two-person games," Journal of Behavioral Decision Making, 7(4), 261–278.
Burks, Stephen V., Carpenter, Je¤rey P., Götte, Lorenz and Rustichini, Aldo (2009):
"Cognitive Skills Explain Economic Preferences, Strategic Behavior, and Job Attachment,"
Proceedings of the National Academy of Sciences, 106(19), 7745–7750.
Burnham, Terence C., Cesarini, David, Johannesson, Magnus, Lichtenstein, Paul and Wallace, Björn (2009): "Higher cognitive ability is associated with lower entries in a p-beauty
contest," Journal of Economic Behavior and Organization, 72, 171–175.
Buzzacchi, Luigi and Pedrini, Stefano (2014): "Does player specialization predict player
actions? Evidence from penalty kicks at FIFA World Cup and UEFA Euro Cup," Applied
Economics, 46(10), 1067–1080.
25
Cappelletti, Dominique, Güth, Werner, and Ploner, Matteo (2011): "Being of two minds:
Ultimatum o¤ers under cognitive constraints," Journal of Economic Psychology, 32(6), 940–
950.
Carpenter, Je¤rey, Graham, Michael and Wolf, Jesse (2013): "Cognitive Ability and
Strategic Sophistication," Games and Economic Behavior, 80(1), 115–130.
Chen, Chun-Ting, Huang, Chen-Ying and Wang, Joseph Tao-yi (2013): "A Window of
Cognition: Eyetracking the Reasoning Process in Spatial Beauty Contest Games," working
paper, National Taiwan University.
Chen, Fadong and Fischbacher, Urs (2015): "Cognitive Processes of Distributional Preferences: A Response Time Study," Working paper, University of Konstanz.
Chiappori, P-A., Levitt, Steven, and Groseclose, Timothy (2002): "Testing mixed-strategy
equilibria when players are heterogeneous: the case of penalty kicks in soccer," American
Economic Review, 92(4), 1138–1151.
Coloma, Germán (2007): "Penalty Kicks in Soccer An Alternative Methodology for Testing
Mixed-Strategy Equilibria," Journal of Sports Economics, 8(5), 530–545.
Corgnet, Brice, Espín, Antonio M., Hernán-González, Roberto, Kujal, Praveen, and Rassenti,
Stephen (2016): "To trust, or not to trust: Cognitive re‡ection in trust games," Journal of
Behavioral and Experimental Economics, 64, 20–27.
Coricelli, Giorgio (2005): "Strategic interaction in iterated zero-sum games," working paper, University of Southern California.
Coricelli, Giorgio and Nagel, Rosemarie (2009): "Neural correlates of depth of strategic
reasoning in medial prefrontal cortex," Proceeings of the National Academy of Science, 106
(23), 9163–9168.
Cornelissen, Gert, Dewitte, Siegfried, and Warlop, Luk (2011): "Are Social Value Orientations expressed automatically? Decision making in the dictator game," Personality and Social
Psychology Bulletin, 37(8), 1080–1090.
Deck, Cary and Jahedi, Salar (2015): "The E¤ect of Cognitive Load on Economic Decision
Making: A Survey and New Experiments," European Economic Review, 78, 97–119.
Devetag, Giovanna and Warglien, Massimo (2003): "Games and phone numbers: Do shortterm memory bounds a¤ect strategic behavior?" Journal of Economic Psychology, 24, 189–202.
Du¤y, Sean, Miller, Dillon, Naddeo, Joseph, Owens, David, and Smith, John (2017): "Are
digit ratios (2D:4D and rel2) related to strategic behavior, academic performance, or the
e¢ cacy of the cognitive load manipulation?" working paper, Rutgers University-Camden.
Du¤y, Sean and Smith, John (2014): "Cognitive Load in the Multi-player Prisoner’s
Dilemma Game: Are There Brains in Games?" Journal of Behavioral and Experimental Economics, 51, 47–56.
26
Emara, Noha, Owens, David, Smith, John, and Wilmer, Lisa (2017): "Serial correlation
and its e¤ects on outcomes in the National Football League," Journal of Behavioral and
Experimental Economics, forthcoming.
Epley, Nicholas and Gilovich, Thomas (2006): "The anchoring-and-adjustment heuristic:
Why the adjustments are insu¢ cient," Psychological Science, 17(4), 311–318.
Fehr, Dietmar and Huck, Ste¤en (2016): "Who knows it is a game? On strategic awareness
and cognitive ability," Experimental Economics, 19(4), 713–726.
Fischbacher, Urs (2007): "z-Tree: Zurich Toolbox for Ready-made Economic Experiments," Experimental Economics, 10(2), 171–178.
Fox, John (1972): "The learning of strategies in a simple, two-person zero-sum game
without saddlepoint," Behavioral Science, 17(3), 300–308.
Geng, Sen, Peng, Yujia, Shachat, Jason, and Zhong, Huizhen (2015): "Adolescents, Cognitive Ability, and Minimax Play," Economics Letters, 128, 54–58.
Georganas, Sotiris, Healy, Paul J., and Weber, Roberto A. (2015): "On the Persistence of
Strategic Sophistication," Journal of Economic Theory, 159(A), 369–400.
Gibbons, Jean Dickinson and Chakraborti, Subhabrata (1992): Nonparametric Statistical
Inference, Marcel Dekker, New York.
Gilbert, Daniel T., Pelham, Brett W., and Krull, Douglas S. (1988): "On Cognitive Busyness: When Person Perceivers Meet Persons Perceived," Journal of Personality and Social
Psychology, 54(5), 733–740.
Gill, David and Prowse, Victoria (2016): "Cognitive Ability, Character Skills, and Learning
to Play Equilibrium: A Level-k Analysis," Journal of Political Economy, 126(4), 1619–1676.
Greiner, Ben (2015): "Subject Pool Recruitment Procedures: Organizing Experiments
with ORSEE," Journal of the Economic Science Association, 1(1), 114–125.
Grimm, Veronika, and Mengel, Friederike (2012): "An experiment on learning in a multiple
games environment," Journal of Economic Theory, 147(6), 2220–2259.
Hauge, Karen Evelyn, Brekke, Kjell Arne, Johansson, Lars-Olof, Johansson-Stenman, Olof,
and Svedsäter, Henrik (2016): "Keeping others in our mind or in our heart? Distribution
games under cognitive load," Experimental Economics, 19(3), 562–576.
Hinson, John M., Jameson, Tina L., and Whitney, Paul (2002): "Somatic markers, working
memory, and decision making," Cognitive, A¤ ective, and Behavioral Neuroscience, 2 (4), 341–
353.
Hinson, John M., Jameson, Tina L., and Whitney, Paul (2003): "Impulsive Decision
Making and Working Memory," Journal of Experimental Psychology: Learning, Memory, and
Cognition, 29(2), 298–306.
27
Hsu, Shih-Hsun, Huang, Chen-Ying, and Tang, Cheng-Tao (2007): "Minimax play at
Wimbledon: Comment," American Economic Review, 97(1), 517–523.
Imhof, Lorens A., Fudenberg, Drew and Nowak, Martin A. (2007): "Tit-for-tat or win-stay,
lose-shift?" Journal of Theoretical Biology, 247(3), 574–580.
Jones, Garett (2008): "Are smarter groups more cooperative? Evidence from prisoner’s
dilemma experiments, 1959-2003," Journal of Economic Behavior and Organization, 68, 489–
497.
Jones, Matthew T. (2014): "Strategic Complexity and Cooperation: An Experimental
Study," Journal of Economic Behavior and Organization, 106, 352–366.
Kiss, H. J., Rodriguez-Lara, I., and Rosa-García, A. (2016): "Think Twice Before Running!
Bank Runs and Cognitive Abilities," Journal of Behavioral and Experimental Economics, 64,
12–19.
Kovash, Kenneth and Levitt, Steven D. (2009): "Professionals do not play minimax: evidence from major League Baseball and the National Football League," working paper, National
Bureau of Economic Research.
Levitt, Steven D., List, John A., and Reiley, David H. (2010): "What happens in the …eld
stays in the …eld: Exploring whether professionals play minimax in laboratory experiments,"
Econometrica, 78(4), 1413–1434.
Lohse, Johannes (2016): "Smart or Sel…sh-When Smart Guys Finish Nice," Journal of
Behavioral and Experimental Economics, 64, 28–40.
Mann, Traci and Ward, Andrew (2007): "Attention, Self-Control, and Health Behaviors,"
Current Directions in Psychological Science, 16(5), 280–283.
Martin, Christopher Flynn, Bhui, Rahul, Bossaerts, Peter, Matsuzawa, Tetsuro, and
Camerer, Colin (2014): "Chimpanzee choice rates in competitive games match equilibrium
game theory predictions," Scienti…c Reports, 4, 5182.
McGarrity, Joseph P. and Linnen, Brian (2010): "Pass or Run: An Empirical Test of the
Matching Pennies Game Using Data from the National Football League," Southern Economic
Journal, 3, 791–810.
Messick, David M. (1967): "Interdependent decision strategies in zero-sum games: A
computer-controlled study," Behavioral Science, 12(1), 33–48.
Milinski, Manfred, and Wedekind, Claus (1998): "Working memory constrains human
cooperation in the Prisoner’s Dilemma," Proceedings of the National Academy of Sciences
95(23), 13755–13758.
Milosavljevic, Milica, Navalpakkam, Vidhya, Koch, Christof, and Rangel, Antonio (2012):
"Relative visual saliency di¤erences induce sizable bias in consumer choice," Journal of Consumer Psychology, 22(1), 67–74.
28
Mo¤att, Peter G. (2005): "Stochastic choice and the allocation of cognitive e¤ort," Experimental Economics, 8(4), 369–388.
Mookherjee, Dilip and Sopher, Barry (1997): "Learning and decision costs in experimental
constant sum games," Games and Economic Behavior, 19(1), 97–132.
Mookherjee, Dilip and Sopher, Barry (1994): "Learning Behavior in an Experimental
Matching Pennies Game," Games and Economic Behavior, 7(1), 62–91.
Morey, Candice C. and Bieler, Malte (2013): "Visual short-term memory always requires
general attention," Psychonomic Bulletin and Review, 20(1), 163–170.
Morey, Candice C. and Cowan, Nelson (2004): "When visual and verbal memories compete:
Evidence of cross-domain limits in working memory," Psychonomic Bulletin and Review, 11(2),
296–301.
Ochs, Jack (1995): "Games with unique, mixed strategy equilibria: An experimental
study," Games and Economic Behavior, 10(1), 202–217.
O’Neill, Barry (1987): "Nonmetric test of the minimax theory of two-person zerosum
games," Proceedings of the National Academy of Sciences, 84(7), 2106–2109.
O’Neill, Barry (1991): "Comments on Brown and Rosenthal’s reexamination," Econometrica, 59(2), 503–507.
Oskarsson, An T., Van Boven, Leaf, McClelland, Gary H., and Hastie, Reid (2009):
"What’s next? Judging sequences of binary events," Psychological Bulletin, 135(2), 262–285.
Palacios-Huerta, Ignacio (2003a): "Professionals play minimax," Review of Economic Studies, 70(2), 395–415.
Palacios-Huerta, Ignacio (2003b) "Learning to Open Monty Hall’s Doors," Experimental
Economics, 6(3), 235–251.
Palacios-Huerta, Ignacio, Olivero, Antonio, Bestmann, Sven, Vila, Jose Florensa, and
Apesteguia, Jose (2014): "Mapping Minimax in the Brain," Beautiful Game Theory: How
Soccer Can Help Economics, Palacios-Huerta, Ignacio, Princeton University Press, Princeton,
New Jersey, 58–67.
Palacios-Huerta, Ignacio and Volij, Oscar (2008): "Experientia docet: Professionals play
minimax in laboratory experiments," Econometrica, 76(1), 71–115.
Proto, Eugenio, Rustichini, Aldo, and So…anos, Andis (2014): "Higher Intelligence Groups
Have Higher Cooperation Rates in the Repeated Prisoner’s Dilemma," working paper, University of Warwick.
Putterman, Louis, Tyran, Jean-Robert and Kamei, Kenju (2011): "Public goods and
voting on formal sanction schemes," Journal of Public Economics, 95, 1213–1222.
29
Rabin, Matthew (2002): "Inference by Believers in the Law of Small Numbers," Quarterly
Journal of Economics, 117(3), 775–816.
Rapoport, Amnon and Amaldoss, Wilfred (2000): "Mixed strategies and iterative elimination of strongly dominated strategies: an experimental investigation of states of knowledge,"
Journal of Economic Behavior and Organization, 42(4), 483–521.
Rapoport, Amnon and Amaldoss, Wilfred (2004): "Mixed-strategy play in single-stage
…rst-price all-pay auctions with symmetric players," Journal of Economic Behavior and Organization, 54(4), 585–607.
Rapoport, Amnon and Boebel, Richard B. (1992): "Mixed strategies in strictly competitive
games: a further test of the minimax hypothesis," Games and Economic Behavior, 4(2), 261–
283.
Rapoport, Amnon and Budescu, David V. (1992): "Generation of random series in twoperson strictly competitive games," Journal of Experimental Psychology: General, 121(3),
352–363.
Raven, John C., De Lemos, Marion M., 1990. Standard Progressive Matrices. Oxford,
Psychologists Press, Oxford, United Kingdom.
Reed, Derek D., Critch…eld, Thomas S., and Martens, Brian K. (2006): "The generalized
matching law in elite sport competition: Football play calling as operant choice," Journal of
Applied Behavior Analysis, 39(3), 281–297.
Roch, Sylvia G., Lane, John A. S., Samuelson, Charles D., Allison, Scott T. and Dent,
Jennifer L. (2000): "Cognitive Load and the Equality Heuristic: A Two-Stage Model of
Resource Overconsumption in Small Groups," Organizational Behavior and Human Decision
Processes, 83(2), 185–212.
Rosenthal, Robert W., Shachat, Jason and Walker, Mark (2003): "Hide and seek in Arizona," International Journal of Game Theory, 32(2), 273–293.
Rubinstein, Ariel (2007): "Instinctive and cognitive reasoning: A study of response times,"
Economic Journal, 117(523), 1243–1259.
Rydval, Ondrej (2011): "The Causal E¤ect of Cognitive Abilities on Economic Behavior:
Evidence from a Forecasting Task with Varying Cognitive Load," working paper, CERGE-EI.
Rydval, Ondrej and Ortmann, Andreas (2004): "How …nancial incentives and cognitive
abilities a¤ect task performance in laboratory settings: an illustration," Economics Letters,
85, 315–320.
Samson, Katarzyna and Kostyszyn, Patrycjusz (2015): "E¤ects of Cognitive Load on
Trusting Behavior –An Experiment Using the Trust Game," PLoS ONE, 10(5), 0127680.
30
Schulz, Jonathan F., Fischbacher, Urs, Thöni, Christian, Utikal, Verena (2014): "A¤ect
and fairness: Dictator games under cognitive load," Journal of Economic Psychology, 41,
77–87.
Schnusenberg, Oliver and Gallo, Andrés (2011): "On cognitive ability and learning in a
beauty contest," Journal for Economic Educators, 11(1), 13–24.
Shachat, Jason M. (2002): "Mixed strategy play and the minimax hypothesis," Journal of
Economic Theory, 104(1), 189–226.
Shachat, Jason and Swarthout, J. Todd (2004): "Do we detect and exploit mixed strategy
play by opponents?" Mathematical Methods of Operations Research, 59(3), 359–373.
Shachat, Jason and Swarthout, J. Todd (2012): "Learning about learning in games through
experimental control of strategic interdependence," Journal of Economic Dynamics and Control, 36(3), 383–402.
Shachat, Jason, Swarthout, J. Todd, and Wei, Lijia (2015): "A hidden Markov model for
the detection of pure and mixed strategy play in games." Econometric Theory, 31(4), 729–752.
Shiv, Baba and Fedorikhin, Alexander (1999): "Heart and Mind in Con‡ict: The Interplay
of A¤ect and Cognition in Consumer Decision Making," Journal of Consumer Research, 26(3),
278–292.
Spiliopoulos, Leonidas (2012): "Pattern recognition and subjective belief learning in a
repeated constant-sum game," Games and Economic Behavior, 75(2), 921–935.
Spiliopoulos, Leonidas (2013): "Strategic adaptation of humans playing computer algorithms in a repeated constant-sum game," Autonomous Agents and Multi-Agent Systems,
27(1), 131–160.
Swann, William B., Hixon, Gregory, Stein-Seroussi, Alan, and Gilbert, Daniel T. (1990):
"The Fleeting Gleam of Praise: Cognitive Processes Underlying Behavioral Reactions to SelfRelevant Feedback," Journal of Personality and Social Psychology, 59 (1), 17–26.
Van den Bos, Kees, Peters, Susanne L., Bobocel, D. Ramona, and Ybema, Jan Fekke
(2006): "On preferences and doing the right thing: Satisfaction with advantageous inequity
when cognitive processing is limited," Journal of Experimental Social Psychology, 42, 273–289.
van’t Veer, Anna E., Stel, Mariëlle, and van Beest, Ilja (2014): "Limited capacity to lie:
Cognitive load interferes with being dishonest," Judgment and Decision Making, 9(3), 199–206.
Van Essen, Matt, and Wooders, John (2015): "Blind Stealing: Experience and Expertise
in a Mixed-Strategy Poker Experiment," Games and Economic Behavior, 91, 186–206.
Wagenaar, Willem A. (1972): "Generation of random sequences by human subjects: A
critical survey of literature," Psychological Bulletin, 77(1), 65–72.
31
Walker, Mark and Wooders, John (2001): "Minimax play at Wimbledon," American Economic Review, 91(5), 1521–1538.
Wang, Zhijian, and Xu, Bin (2014): "Incentive and stability in the Rock-Paper-Scissors
game: an experimental investigation," working paper, arXiv preprint arXiv:1407.1170.
Wang, Zhijian, Xu, Bin, and Zhou, Hai-Jun (2014): "Social cycling and conditional responses in the Rock-Paper-Scissors game," Scienti…c Reports, 4, 5830.
Ward, Andrew and Mann, Traci (2000): "Don’t Mind If I Do: Disinhibited Eating Under
Cognitive Load," Journal of Personality and Social Psychology, 78 (4), 753–763.
Whitney, Paul, Rinehart, Christa A., and Hinson, John M. (2008): "Framing e¤ects under
cognitive load: The role of working memory in risky decisions," Psychonomic Bulletin and
Review, 15(6), 1179–1184.
Wilcox, Nathaniel T. (1993): "Lottery choice: Incentives, complexity and decision time,"
Economic Journal, 103(421), 1397–1417.
Wooders, John (2010): "Does experience teach? Professionals and minimax play in the
lab," Econometrica, 78(3), 1143–1154.
Zokaei, Nahid, Heider, Maike, and Husain, Masud (2014): "Attention is required for maintenance of feature binding in visual working memory," Quarterly Journal of Experimental
Psychology, 67(6), 1191–1213.
32
Figure 2: Kolmogorov-Smirnov Test of Runs for subjects under a low load.
33
Figure 3: Kolmogorov-Smirnov Test of Runs for subjects under a high load.
34
Supplemental Online Appendix
Here we summarize regression (2) of Table 6 but restricted by computer opponent treatment.
The speci…cation includes repeated measures and is summarized in Table A1.
Table A1: Regressions of earnings across rounds
Naive 50-50 Naive Pattern
High load
0:099
0:0464
(0:061)
(0:0786)
Round
0:0025y
0:0017
(0:0014)
(0:0012)
Round*High load
0:0045
0:0022
(0:0021)
(0:0019)
AIC
8850:3
6922:2
Observations
3450
3050
Exploitative WSLS
0:0577
(0:0622)
0:0018
(0:0015)
0:0034y
(0:0020)
7825:3
3200
Exploitative Mix
0:0848
(0:0566)
0:0014
(0:0014)
0:0009
(0:0018)
7481:5
3300
The regressions estimate an exchangeable covariance matrix, clustered by subject. We do not provide the estimates of the intercepts or the covariance estimates. AIC refers to the Akaike information criterion (Akaike, 1974).
denotes
y
p < 0:001, denotes p < 0:01, denotes p < 0:05, and denotes p < 0:1.
35
Screenshot of game
36
Low Load against Exploitative opponent
2
Subject #Down
p-value Runs F (r)
F (r 1) U draw
2
22
2:56
0:110
25
0:483
0:371
0:387
3
22
2:56
0:110
31
0:956
0:921
0:939
4
25
6:25
0:012
21
0:099
0:058
0:058
6
23
3:61
0:057
19
0:033
0:017
0:031
9
28
11:56 < 0:001
32
0:977
0:956
0:962
13
27
9:61
0:002
26
0:576
0:460
0:506
16
28
11:56 < 0:001
20
0:068
0:037
0:060
20
22
2:56
0:110
42
0:999
0:999
0:999
22
49
94:09 < 0:001
3
1:000
0:040
0:202
27
24
4:84
0:028
37
0:999
0:999
0:999
30
22
2:56
0:110
34
0:995
0:989
0:990
31
29
13:69 < 0:001
33
0:992
0:983
0:989
33
25
6:25
0:012
32
0:969
0:942
0:955
34
18
0:16
0:689
29
0:959
0:916
0:934
43
31
18:49 < 0:001
26
0:718
0:612
0:650
47
33
24:01 < 0:001
16
0:014
0:006
0:012
49
27
9:61
0:002
34
0:994
0:987
0:990
52
22
2:56
0:110
28
0:796
0:704
0:714
53
27
9:61
0:002
24
0:351
0:250
0:286
62
25
6:25
0:0124
30
0:902
0:841
0:872
64
17
0:01
0:9203
21
0:270
0:172
0:218
65
32
21:16 < 0:001
24
0:550
0:434
0:497
67
50
1
0
0
0
68
29
13:69 < 0:001
26
0:629
0:516
0:556
70
24
4:84
0:028
19
0:0314
0:016
0:020
71
28
11:56 < 0:001
30
0:921
0:869
0:902
77
27
9:61
0:002
31
0:949
0:911
0:920
78
33
24:01 < 0:001
15
0:0061
0:002
0:003
102
30
16:00 < 0:001
25
0:559
0:438
0:535
Camden subjects are labeled 1 78. Haverford subjects are labeled 101 152. Among the
50 decisions against an exploitative opponent, we report the number of down actions, the 2
statistic and the corresponding p-value as discussed in subsection 4:5. We report the number
of runs, the two CDF statistics, and the draw of the uniform between these, as discussed in
subsection 4:6.
37
Low Load against Exploitative opponent
2
Subject #Down
p-value Runs F (r) F (r 1) U draw
103
34
27:04 < 0:001
21
0:341
0:223
0:326
105
48
88:36 < 0:001
4
0:118
0:041
0:075
110
25
6:25
0:012
21
0:098
0:058
0:067
112
35
30:25 < 0:001
26
0:936
0:892
0:932
119
28
11:56 < 0:001
26
0:598
0:483
0:575
120
25
6:25
0:012
26
0:558
0:442
0:467
121
34
27:04 < 0:001
16
0:021
0:009
0:017
125
27
9:61
0:002
36
0:999
0:998
0:998
128
36
33:64 < 0:001
23
0:805
0:663
0:793
130
30
16:00 < 0:001
25
0:559
0:438
0:480
131
25
6:25
0:012
25
0:442
0:335
0:388
134
22
2:56
0:110
28
0:796
0:704
0:791
136
25
6:25
0:012
22
0:159
0:098
0:149
137
27
9:61
0:002
17
0:008
0:003
0:007
140
41
53:29 < 0:001
14
0:237
0:151
0:195
141
24
4:84
0:028
25
0:447
0:339
0:399
142
25
6:25
0:012
28
0:763
0:665
0:683
144
23
3:61
0:057
20
0:062
0:033
0:052
145
28
11:56 < 0:001
26
0:598
0:483
0:561
146
25
6:25
0:012
29
0:841
0:763
0:818
147
24
4:84
0:028
24
0:339
0:240
0:270
149
36
33:64 < 0:001
19
0:282
0:167
0:219
150
25
6:25
0:012
27
0:665
0:558
0:641
151
30
16:00 < 0:001
29
0:912
0:850
0:889
152
23
3:61
0:057
33
0:987
0:973
0:973
Camden subjects are labeled 1 78. Haverford subjects are labeled 101 152. Among the
50 decisions against an exploitative opponent, we report the number of down actions, the 2
statistic and the corresponding p-value as discussed in subsection 4:5. We report the number
of runs, the two CDF statistics, and the draw of the uniform between these, as discussed in
subsection 4:6.
38
High Load against Exploitative opponent
2
Subject #Down
p-value Runs F (r) F (r 1) U draw
1
36
33:64 < 0:001
19
0:282
0:167
0:189
5
24
4:84
0:028
28
0:767
0:669
0:716
7
26
7:84
0:005
28
0:767
0:669
0:743
8
28
11:56 < 0:001
26
0:598
0:483
0:501
10
9
5:29
0:021
17
0:829
0:576
0:613
11
31
18:49 < 0:001
19
0:063
0:033
0:059
12
31
18:49 < 0:001
21
0:177
0:109
0:122
14
22
2:56
0:110
32
0:977
0:956
0:974
15
30
16:00 < 0:001
19
0:051
0:026
0:045
17
23
3:61
0:057
31
0:949
0:911
0:925
18
22
2:56
0:110
16
0:004
0:001
0:003
19
20
1:00
0:317
33
0:995
0:988
0:990
21
40
49:00 < 0:001
16
0:368
0:260
0:349
23
35
30:25 < 0:001
23
0:699
0:551
0:581
24
32
21:16 < 0:001
17
0:021
0:001
0:014
25
37
37:21 < 0:001
24
0:942
0:902
0:916
26
29
13:69 < 0:001
32
0:983
0:966
0:983
28
38
40:96 < 0:001
17
0:254
0:135
0:208
29
21
1:69
0:194
25
0:516
0:399
0:496
32
32
21:16 < 0:001
28
0:916
0:862
0:913
35
25
6:25
0:012
23
0:237
0:159
0:163
36
26
7:84
0:005
37
0:999
0:998
0:999
37
20
1:00
0:317
30
0:950
0:912
0:947
38
28
11:56 < 0:001
25
0:483
0:371
0:440
39
28
11:56 < 0:001
30
0:921
0:869
0:869
Camden subjects are labeled 1 78. Haverford subjects are labeled 101 152. Among the
50 decisions against an exploitative opponent, we report the number of down actions, the 2
statistic and the corresponding p-value as discussed in subsection 4:5. We report the number
of runs, the two CDF statistics, and the draw of the uniform between these, as discussed in
subsection 4:6.
39
High Load against Exploitative opponent
2
Subject #Down
p-value Runs F (r) F (r 1) U draw
40
23
3:61
0:057
23
0:250
0:169
0:193
41
24
4:84
0:028
27
0:669
0:562
0:610
42
21
1:69
0:194
27
0:735
0:629
0:673
44
49
94:09 < 0:001
3
1
0:04
0:929
45
18
0:16
0:689
26
0:772
0:676
0:676
46
19
0:49
0:484
22
0:264
0:177
0:185
48
28
11:56 < 0:001
34
0:995
0:989
0:990
50
31
18:49 < 0:001
16
0:007
0:003
0:005
51
33
24:01 < 0:001
21
0:270
0:172
0:230
54
31
18:49 < 0:001
25
0:612
0:488
0:539
55
22
2:56
0:110
27
0:704
0:598
0:655
56
29
13:69 < 0:001
20
0:077
0:043
0:077
57
23
3:61
0:057
27
0:682
0:576
0:659
58
23
3:61
0:057
22
0:169
0:105
0:120
59
25
6:25
0:012
32
0:969
0:942
0:948
60
28
11:56 < 0:001
29
0:869
0:796
0:851
61
32
21:16 < 0:001
21
0:217
0:135
0:193
63
23
3:61
0:057
28
0:778
0:682
0:744
66
30
16:00 < 0:001
30
0:950
0:912
0:946
69
23
3:61
0:057
25
0:460
0:351
0:368
72
24
4:84
0:028
33
0:985
0:970
0:984
73
20
1:00
0:317
20
0:090
0:051
0:051
74
22
2:56
0:110
36
0:999
0:998
0:999
75
33
24:01 < 0:001
7
0:000
0:000
0:000
76
49
94:09 < 0:001
3
1
0:04
0:296
Camden subjects are labeled 1 78. Haverford subjects are labeled 101 152. Among the
50 decisions against an exploitative opponent, we report the number of down actions, the 2
statistic and the corresponding p-value as discussed in subsection 4:5. We report the number
of runs, the two CDF statistics, and the draw of the uniform between these, as discussed in
subsection 4:6.
40
High Load against Exploitative opponent
2
Subject #Down
p-value Runs F (r) F (r 1) U draw
101
31
18:49 < 0:001
30
0:965
0:936
0:951
104
31
18:49 < 0:001
27
0:815
0:718
0:727
106
27
9:61
0:002
35
0:998
0:994
0:997
107
28
11:56 < 0:001
31
0:956
0:921
0:952
108
24
4:84
0:028
24
0:339
0:240
0:271
109
23
3:61
0:057
23
0:250
0:169
0:224
111
21
1:69
0:194
26
0:629
0:516
0:549
113
24
4:84
0:028
29
0:844
0:767
0:816
114
20
1:00
0:317
31
0:975
0:950
0:952
115
20
1:00
0:317
24
0:438
0:327
0:372
116
24
4:84
0:028
29
0:844
0:767
0:832
117
24
4:84
0:028
33
0:985
0:970
0:984
118
24
4:84
0:028
32
0:970
0:944
0:953
122
19
0:49
0:484
16
0:007
0:003
0:004
123
29
13:69 < 0:001
18
0:022
0:010
0:017
124
27
9:61
0:002
27
0:682
0:576
0:612
126
24
4:84
0:028
30
0:904
0:844
0:861
127
30
16:00 < 0:001
31
0:975
0:950
0:958
129
42
57:76 < 0:001
17
1
0:822
0:937
132
24
4:84
0:028
40
0:999
0:999
0:999
133
29
13:69 < 0:001
30
0:935
0:889
0:891
135
24
4:84
0:028
30
0:904
0:844
0:858
138
26
7:84
0:005
28
0:767
0:669
0:691
139
26
7:84
0:005
30
0:904
0:844
0:889
143
26
7:84
0:005
21
0:100
0:059
0:071
148
31
18:49 < 0:001
29
0:936
0:883
0:916
Camden subjects are labeled 1 78. Haverford subjects are labeled 101 152. Among the
50 decisions against an exploitative opponent, we report the number of down actions, the 2
statistic and the corresponding p-value as discussed in subsection 4:5. We report the number
of runs, the two CDF statistics, and the draw of the uniform between these, as discussed in
subsection 4:6.
41